Multiply the Matrices

Problem

[[19,−21,3,4],[−5,8,−1,−10],[−1,−5,−1,6],[−1,14,9,3]]*[[19,−5,−1,−1],[−21,8,−5,14],[3,−1,−1,9],[4,−10,6,3]]

Solution

1. Identify the dimensions of the matrices. Both are 4×4 matrices, so the resulting product will also be a 4×4 matrix.

2. Calculate the elements of the first row by taking the dot product of the first row of the left matrix with each column of the right matrix.

(R_1)*(C_1)=(19)*(19)+(−21)*(−21)+(3)*(3)+(4)*(4)=361+441+9+16=827

(R_1)*(C_2)=(19)*(−5)+(−21)*(8)+(3)*(−1)+(4)*(−10)=−95−168−3−40=−306

(R_1)*(C_3)=(19)*(−1)+(−21)*(−5)+(3)*(−1)+(4)*(6)=−19+105−3+24=107

(R_1)*(C_4)=(19)*(−1)+(−21)*(14)+(3)*(9)+(4)*(3)=−19−294+27+12=−274

3. Calculate the elements of the second row.

(R_2)*(C_1)=(−5)*(19)+(8)*(−21)+(−1)*(3)+(−10)*(4)=−95−168−3−40=−306

(R_2)*(C_2)=(−5)*(−5)+(8)*(8)+(−1)*(−1)+(−10)*(−10)=25+64+1+100=190

(R_2)*(C_3)=(−5)*(−1)+(8)*(−5)+(−1)*(−1)+(−10)*(6)=5−40+1−60=−94

(R_2)*(C_4)=(−5)*(−1)+(8)*(14)+(−1)*(9)+(−10)*(3)=5+112−9−30=78

4. Calculate the elements of the third row.

(R_3)*(C_1)=(−1)*(19)+(−5)*(−21)+(−1)*(3)+(6)*(4)=−19+105−3+24=107

(R_3)*(C_2)=(−1)*(−5)+(−5)*(8)+(−1)*(−1)+(6)*(−10)=5−40+1−60=−94

(R_3)*(C_3)=(−1)*(−1)+(−5)*(−5)+(−1)*(−1)+(6)*(6)=1+25+1+36=63

(R_3)*(C_4)=(−1)*(−1)+(−5)*(14)+(−1)*(9)+(6)*(3)=1−70−9+18=−60

5. Calculate the elements of the fourth row.

(R_4)*(C_1)=(−1)*(19)+(14)*(−21)+(9)*(3)+(3)*(4)=−19−294+27+12=−274

(R_4)*(C_2)=(−1)*(−5)+(14)*(8)+(9)*(−1)+(3)*(−10)=5+112−9−30=78

(R_4)*(C_3)=(−1)*(−1)+(14)*(−5)+(9)*(−1)+(3)*(6)=1−70−9+18=−60

(R_4)*(C_4)=(−1)*(−1)+(14)*(14)+(9)*(9)+(3)*(3)=1+196+81+9=287

Final Answer

[[19,−21,3,4],[−5,8,−1,−10],[−1,−5,−1,6],[−1,14,9,3]]*[[19,−5,−1,−1],[−21,8,−5,14],[3,−1,−1,9],[4,−10,6,3]]=[[827,−306,107,−274],[−306,190,−94,78],[107,−94,63,−60],[−274,78,−60,287]]

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